Question

Найдите разность между наименьшим положительным и наибольшем отрицательным корням уравнения sin^3 x-6 cosП/6 * cos^3=0 ответ укажите в градусах.

Answer 1

Sin³x-6*√3/2cos³x=0
sin³x-3√3cos³x=0
(sinx-√3cosx)(sin²x+√3sinxcosx+3cos²x)=0
sinx-√3cosx=0
2(1/2*sinx-√3/2cosx)=0
sin(x-π/3)=0
x-π/3=πn,n∈z
x=π/3+πn,n∈z
sin²x+√3sinxcosx+3cos²x=0/cos²x
tg²x+√3tgx+3=0
tgx=a
a²+√3a+3=0
D=3-12=-9<0 нет решения
x1=60 наим полож
х2=-300 наиб отр
60-(-300)= 360

Answer 2

Calculation of the Difference Between the Smallest Positive and Largest Negative Roots of the Equation

To find the difference between the smallest positive and largest negative roots of the equation sin^3(x) - 6cos(π/6) * cos^3(x) = 0, we need to solve the equation and find the values of x that satisfy it.

Let's break down the steps to solve the equation:

1. Start by simplifying the equation using trigonometric identities: - sin^3(x) - 6cos(π/6) * cos^3(x) = 0 - Since cos(π/6) = √3/2, the equation becomes: sin^3(x) - 6(√3/2) * cos^3(x) = 0 - Simplifying further: sin^3(x) - (3√3) * cos^3(x) = 0

2. Apply the identity sin^2(x) + cos^2(x) = 1 to rewrite the equation: - sin^3(x) - (3√3) * (1 - sin^2(x)) * cos(x) = 0 - Simplifying: sin^3(x) - (3√3) * (cos(x) - sin^2(x) * cos(x)) = 0 - Expanding: sin^3(x) - (3√3) * cos(x) + (3√3) * sin^2(x) * cos(x) = 0

3. Factor out cos(x) from the equation: - cos(x) * (sin^2(x) + (3√3) * sin^2(x) - (3√3)) = 0 - Simplifying: cos(x) * (4√3 * sin^2(x) - (3√3)) = 0

4. Set each factor equal to zero and solve for x: - First factor: cos(x) = 0 - The solutions for this factor occur when x is equal to π/2 + nπ, where n is an integer. - Second factor: 4√3 * sin^2(x) - (3√3) = 0 - Simplifying: 4√3 * sin^2(x) = 3√3 - Dividing both sides by 4√3: sin^2(x) = 3/4 - Taking the square root of both sides: sin(x) = ±√(3/4) - Since sin(x) is positive in the first and second quadrants, the solutions for this factor occur when x is equal to arcsin(√(3/4)) + 2nπ or π - arcsin(√(3/4)) + 2nπ, where n is an integer.

5. Now that we have the possible values of x, we can find the smallest positive and largest negative roots and calculate their difference.

Let's calculate the values of x and find the difference between the smallest positive and largest negative roots.

- For the factor cos(x) = 0, the solutions are x = π/2 + nπ, where n is an integer. - For the factor sin(x) = √(3/4), the solutions are x = arcsin(√(3/4)) + 2nπ or π - arcsin(√(3/4)) + 2nπ, where n is an integer.

To find the smallest positive root, we need to find the smallest positive value of x that satisfies the equation. The smallest positive root occurs when n = 0.

To find the largest negative root, we need to find the largest negative value of x that satisfies the equation. The largest negative root occurs when n = -1.

Let's calculate the values of x:

- For the factor cos(x) = 0, the smallest positive root is x = π/2. - For the factor sin(x) = √(3/4), the smallest positive root is x = arcsin(√(3/4)).

Now, let's calculate the difference between the smallest positive and largest negative roots:

- Difference = x(smallest positive) - x(largest negative) - Difference = (π/2) - (arcsin(√(3/4)))

To calculate the value of the difference in degrees, we can convert radians to degrees:

- Difference (in degrees) = (π/2) - (arcsin(√(3/4))) * (180/π)

Let's calculate the value of the difference in degrees:

- Difference (in degrees) = (π/2) - (arcsin(√(3/4))) * (180/π)